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A Jacobi Spectral Collocation Scheme Based on Operational Matrix for Time-fractional Modified Korteweg-de Vries Equations

A.H. Bhrawy1,2, E.H. Doha3, S.S. Ezz-Eldien4, M.A. Abdelkawy2
Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia.
Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt.
Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt.
Department of Mathematics, Faculty of Science, Assiut University, New Valley Branch, El-Kharja 72511, Egypt.

Computer Modeling in Engineering & Sciences 2015, 104(3), 185-209. https://doi.org/10.3970/cmes.2015.104.185

Abstract

In this paper, a high accurate numerical approach is investigated for solving the time-fractional linear and nonlinear Korteweg-de Vries (KdV) equations. These equations are the most appropriate and desirable definition for physical modeling. The spectral collocation method and the operational matrix of fractional derivatives are used together with the help of the Gauss-quadrature formula in order to reduce such problem into a problem consists of solving a system of algebraic equations which greatly simplifying the problem. Our approach is based on the shifted Jacobi polynomials and the fractional derivative is described in the sense of Caputo. In addition, the presented approach is applied also to solve the timefractional modified KdV equation. For testing the accuracy, validity and applicability of the developed numerical approach, we apply it to provide high accurate approximate solutions for four test problems.

Keywords

KdV equation, Jacobi polynomials, Operational matrix, Gauss quadrature, Collocation spectral method, Caputo derivative

Bhrawy, A., Doha, E., Ezz-Eldien, S., Abdelkawy, M. (2015). A Jacobi Spectral Collocation Scheme Based on Operational Matrix for Time-fractional Modified Korteweg-de Vries Equations. CMES-Computer Modeling in Engineering & Sciences, 104(3), 185–209.

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